Geometry of Singularities for the Steady Boussinesq Equations

1. I n t r o d u c t i o n The possibility of singularity formation from smooth initial data for the solution of the Euler equations for inviscid, incompressible fluid flow in three dimensions is one of the main unsolved problems of mathematical fluid mechanics. In spite of considerable effort on this problem from analytical, numerical and physical approaches, there is still little convincing evidence either for or against the possibility of singularity formation. * Research supported in part by the ARPA under URI grant number # N00014092-J-1890. t Research supported in part by the NSF under grant number #DMS93-02013. t Research supported in part by the NSF under grant #DMS-9306488. 370 R .E . Caflisch, N. Ercotani and G. Steele Selecta Math. In this paper, singularities are analyzed for a rela*ed, but much simpler, system, the steady Boussinesq equations: u . V p = 0 u . V~ = -p~ V x u = ~ (1.1) V . u = 0 . This system describes steady, two-dimensional, stratified (i.e., variable density), incompressible flow in which p is density, u is two dimensional velocity and ~ is vorticity. The buoyancy term p~, plays the role of vortex stretching in the Euler equations. This system is derived in the "Boussinesq limit," in which the variation of density is important in buoyancy terms but insignificant in inertial terms. For the steady Boussinesq equations (1.1) we will derive the generic tbrm of singularities under certain natural restrictions. Singularities for complex solutions and at, complex spatial positions are considered in this study, but the results are also valid for steady real singularities of real solutions. In addition, we present numerical computations which confirm these generic singularity types. The problem of singularity formation from smooth initial data for the (timedependent) Euler or Boussinesq equations is the motivation for this work. Our reasons for studying complex singularities for steady flows are twofold: First, this problem serves as a vehicle tbr developing methods that may be applicable to singularity formation in the unsteady problem. For example, the present analysis includes solutions with infinite values for velocity and vorticity. Such infinite values in the dependent variables were excluded from earlier approaches [11]. In fact, a new analysis of Legendrian singularities at which the "velocities" are infinite, which we call "Legendrian blowup," is also included in the analysis below. Second, we believe that the form of steady complex singularities may be indicative of the form for dynamic real singularities or for nearly singular flow. One reason for this belief is that the genericity results for steady singularities can also be interpreted as genericity results for complex singularities that move in the complex plane without change in ttmir structure. This is described in Appendix A. On the other hand, we have found that, for generic steady singularities, the density p is infinite. Since, infinite values of p cannot occur for singularity tbrmation from smooth initial data, this would restrict the set of steady singularities that are relevant to singularity formation. Further discussion of these points is presented in Section 8. The main result of this paper can be informally stated as follows: Consider singularities for the steady Boussinesq system (1.1) which have codimension 1 and for which the vorticity ~ is infinite. The generic form of such a singularity is of two types: (1) ¢ ~ za/2; v .~ xl/2; ~ ~ x -~/2 (2) ~) ~ x l /2 ; V ~ x l / 2 ; ~ ~ X -3 /2 . Vol. 2 (1996) Singularities for Steady Boussinesq Equations 371 If in addition the stream function ¢ is real analytic and is assumed to have an additional symmetry (¢(x, 0) = ¢ ( x , 0)), then the singularity type is (3) ¢,-~ X2/3; V ~ X--1/3; ~ ,~ X -4/3. In these formulas x and v denote some suitably chosen space and velocity coordinates. The term "generic" here is used in the technical sense that is standard in singularity theory (see for instance [2]). A precise statement of this result will be presented in Section 3. The principal ingredient in this study will be a geometric approach to differential equations, which has been developed by two of the present authors and their coworkers for simpler systems in [11], as well as by Bryant, Griffiths and Hsu [7], [8] and Rakhimov [23]. In this approach the solution of a differential equation is viewed as a surface in an appropriate jet space (described in Section 5), and the PDE serves as a constraint on the possible form of this surface. In particular, for the steady Boussinesq equations, we show that the convective equations can be solved exactly and that incompressibility provides a constraint on the singularity type, but that the vorticity equations do not lead to any constraint. (For other applications of contact geometry to PDE's the reader may be interested in consulting [1].) The main role of analysis in this approach is to show that such constrained surfaces are generically smooth, which is demonstrated using the Cauchy-Kowalewski Theorem for analytic solutions of a PDE. Geometric and algebraic methods, such as the singularity theory of Arnold [2], can then be used to analyze the generic types of singularities for this surface. The background for this study consists of several analytical results and a numerical computation: First, the mathematical significance of Euler or Boussinesq singularity formation (for the time-dependent equations), should it occur, is that it would limit the validity of mathematical existence theory. Beale, Kato and Majda [6] showed that. if the initial velocity u0 is in Sobolev space H s for some s _> 3, but at time t = t* > 0, u(t) is not in H s, then t* L I1 ( ,t)[I (1.2) in which ]l" I1~ is the L ~ norm in space and w is the vorticity. Bardos and Benachour [5] proved a related, earlier result in an analytic function class; E and Shu [17] proved a similar result for the Boussinesq equations. The physical significance of singularities is less clear and depends critically on the robustness and type of singularities. Singularities might serve as an important means for transfer of energy from large to small scales. In this way they could be responsible for the onset of turbulent flow or even for the continuation of fully developed turbulence. A related analytic result, first stated by Onsager [21] and refined in [12], [16], [18], says that if a weak Euler solution does not conserve energy and it has HSlder exponent a on a set of codimension n, then a + n/3 <_ 1/3. This 372 R.E. Caflisch, N. Ercolani and G. Steele Selecta Math. is physically meaningful, since there is clear evidence that the energy dissipation for the Navier Stokes solution remains nonzero in the zero viscosity limit. The time-dependent Boussinesq equations are